Finite Frame Varieties: Nonsingular Points, Tangent Spaces, and Explicit Local Parameterizations

The (μ,S)-frames are frames with lengths in [μ1⋅⋅⋅μN] and with frame operator S, or the $F=[f_{1}\cdots f_{N}]\in M_{d\times N}(\mathbb{E})$ with column lengths listed by μ and which satisfy FF∗=S. In this paper, we characterize the nonsingular points of real and complex finite (μ,S)-frame varieties by determining where generalized tori and distorted Stiefel manifolds intersect transversally in Hilbert-Schmidt spheres. This provides us with a characterization of the tangent space for each nonsingular point of the (μ,S)-frame varieties, and we leverage this characterization to demonstrate the existence of structured, locally well defined analytic coordinate patches. We conclude by deriving explicit expressions for these coordinates.

[1]  Bernhard G. Bodmann,et al.  Equiangular tight frames from complex Seidel matrices containing cube roots of unity , 2008, 0805.2014.

[2]  V. Pan,et al.  Polynomial and Matrix Computations , 1994, Progress in Theoretical Computer Science.

[3]  Nate Strawn,et al.  Manifold structure of spaces of spherical tight frames , 2003, math/0307367.

[4]  Peter G. Casazza,et al.  Equal-Norm Tight Frames with Erasures , 2003, Adv. Comput. Math..

[5]  R. Duffin,et al.  A class of nonharmonic Fourier series , 1952 .

[6]  I. Daubechies,et al.  PAINLESS NONORTHOGONAL EXPANSIONS , 1986 .

[7]  Deguang Han,et al.  Frames, bases, and group representations , 2000 .

[8]  Mátyás A. Sustik,et al.  On the existence of equiangular tight frames , 2007 .

[9]  John J. Benedetto,et al.  Finite Normalized Tight Frames , 2003, Adv. Comput. Math..

[10]  Robert W. Heath,et al.  Designing structured tight frames via an alternating projection method , 2005, IEEE Transactions on Information Theory.

[11]  Ioan Mackenzie James,et al.  The topology of Stiefel manifolds , 1976 .

[12]  Robert W. Heath,et al.  Generalized Finite Algorithms for Constructing Hermitian Matrices with Prescribed Diagonal and Spectrum , 2005, SIAM J. Matrix Anal. Appl..

[13]  Shijun Zheng,et al.  Harmonic Analysis Related to Schrödinger Operators , 2007, 0711.3262.

[14]  David R. Larson,et al.  Wavelets, frames and operator theory : Focused Research Group Workshop on Wavelets, Frames and Operator Theory, January 15-21, 2003, University of Maryland, College Park, Maryland , 2004 .

[15]  John J. Benedetto,et al.  Sigma-delta quantization and finite frames , 2004, ICASSP.

[16]  Pedro G. Massey,et al.  Minimization of convex functionals over frame operators , 2007, Adv. Comput. Math..

[17]  Thomas Strohmer,et al.  GRASSMANNIAN FRAMES WITH APPLICATIONS TO CODING AND COMMUNICATION , 2003, math/0301135.

[18]  Peter G. Casazza,et al.  Gradient descent of the frame potential , 2009 .

[19]  J. Benedetto,et al.  Second-order Sigma–Delta (ΣΔ) quantization of finite frame expansions , 2006 .

[20]  I. Olkin,et al.  Inequalities: Theory of Majorization and Its Applications , 1980 .

[21]  G. E. Bredon.  Topology and geometry , 1993 .

[22]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[23]  Keri Kornelson,et al.  Ellipsoidal tight frames and projection decompositions of operators , 2003 .

[24]  R. Young,et al.  An introduction to nonharmonic Fourier series , 1980 .

[25]  Nate Strawn.  Geometry and constructions of finite frames , 2009 .

[26]  Thomas Strohmer,et al.  A note on equiangular tight frames , 2008 .

[27]  Vivek K. Goyal,et al.  Quantized Overcomplete Expansions in IRN: Analysis, Synthesis, and Algorithms , 1998, IEEE Trans. Inf. Theory.

[28]  Keri Kornelson,et al.  RANK-ONE DECOMPOSITION OF OPERATORS AND CONSTRUCTION OF FRAMES , 2003 .

[29]  Shayne Waldron,et al.  Generalized Welch bound equality sequences are tight fram , 2003, IEEE Trans. Inf. Theory.

[30]  J. Benedetto,et al.  Sigma-delta (/spl Sigma//spl Delta/) quantization and finite frames , 2006, IEEE Transactions on Information Theory.

[31]  Harold R. Parks,et al.  The Implicit Function Theorem , 2002 .

[32]  Jelena Kovacevic,et al.  Real, tight frames with maximal robustness to erasures , 2005, Data Compression Conference.

[33]  Rayan Saab,et al.  Quantization and Finite Frames , 2013 .